Coordinated 6-DOF Control of Dual Spacecraft Formation

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1 Coordinated 6-DOF Control of Dual Spacecraft Formation M.S.Siva, Scientist, ISRO Satellite Centre, Bangalore, INDIA R.Pandiyan, Scientist, ISRO Satellite Centre, Bangalore, INDIA Debasish Ghose, Professor, Indian Institute of Science, Bangalore, INDIA M. S. Bhat, Professor, Indian Institute of Science, Bangalore, INDIA M.P.Ramachandran, Scientist, ISRO Satellite Centre, Bangalore, INDIA

2 Needs of Formation Flying Degree of performance proportional to instrument size Payload distribution over a no. of satellites High Resolution Wider Swath Simultaneous coverage Modularity Launch Flexibility

3 Investigations by Formation Flying SAR interferometry Planet Finding and Imaging Gravimetry for Gravitational Mapping Stereo Imaging (addressed in this paper) Space Weather Monitoring 3 D Mapping

4 Requirements of Stereo Imaging by Multi Spacecrafts Relative position modeling of formations including perturbations GPS based attitude and Position Estimation for Constellation and Earth Orbit Formations So far researchers concentrated mainly on relative position estimation and control Very Little effort has gone in the field of Relative Attitude Control Position and Attitude Loops were analyzed in decoupled concepts Co ordinated 6 dof control effort using modern control theory

5 Formation Control Requirements Autonomous real time control of relative position and relative attitude among formation members Relative Attitude Maintenance is needed to orient the thrust direction for efficient reconfiguration of position loop Data collection by formation cluster is a function of both spacecraft position and attitude Attenuate the effect of perturbing forces which otherwise disturbs the formation either by inter satellite interference and collisions or by path divergence Performance trade off between Mission Goals and Fuel cost. General guideline is to meet the baseline requirement of x, the control accuracy should be better than x/10 and the navigation or sensing accuracy should be better than x/100.

6 Contributions of the Paper An innovative coupled dynamics and control algorithm is developed for High Precision Stereo Imaging by a dualmicrosatellite formation flying mission 6 Degree of Freedom model where each follower generates the attitude references in real time, based on relative position and translational motion between the leader and its followers For the orbit control loop, the desired parameters are specified in terms of orbital elements and GPS + accelerometer forms the feedback source Desired attitude reference is derived on board from the specified Earth view co ordinates. Star tracker + gyro provides feedback information

7 Orbit Elements Defined in ECI

8 Relative Orbit Elements (ROE) Δα = Δa afδex a F Δe Y a F Δi X a F Δi Y afδu = a L a F a F e L cosωl e F cosωf a ω ω F e L sin L e F sin F a F ( i L i F ) a F af ul uf ( ) ( ) ( ΩL ΩF) sin i F ( ) L: Leader s/c ; F: Follower s/c

9 ROE to Cartesian or Hill s Vector (Position, Velocity) Δr x 1 Δr y 1.5 u u0 Δr z 0 = Δv x 0 Δv y 1.5n Δv z 0 ( ) cosu 2sinu 0 nsinu 2ncosu 0 sinu 2cosu 0 ncosu 2nsinu sinu 0 0 ncosu 0 coti cosu 0 0 nsinu Δα u and u 0 are the mean argument of latitude at any time t and epoch time t 0 n is the mean orbital rate in rad/s

10 Relative Attitude Kinematics The Relative Attitude of the Follower S/C with respect to Leader is given by q = q L* q F

11 Relative Attitude Modeling q& 1 = q 2 ( ω) ω=ω F L,F =ω F I,F +ω F L,I =ω F I,F ω F I,L

12 Navigation System IRAP (Inertial Reference Gyro with Accelerometer Package) with GPS and Star Tracker Updates forms the Navigation Loop Attitude Path Star Tracker provides the instantaneous attitude of the spacecraft with respect to Inertial Frame. Gyro data is used to propagate the attitude between star tracker updates. This provides the instantaneous attitude of the spacecraft with respect to the Inertial Frame which is used along with the gyro rate for the leader. For the follower spacecraft control, the relative attitude and rate of the follower with respect to the leader s orbit reference frame is represented in the follower frame

13 Navigation System (Contd ) Position Path Accelerometer measures the acceleration in body frame which is represented in inertial frame using transformation matrix obtained from the instantaneous body to inertial attitude quaternion. This transformed acceleration is double integrated and used to propagate the position information between absolute GPS updates. The processed Cartesian data is transformed to instantaneous orbital elements and fed to the controller

14 Controller Design Linear Quadratic Controller (LQR) is used for both position and attitude control loops. This is based on optimization of a specified cost functional For the position Loop, fuel consumption is the cost functional as thruster is the prime actuator to maintain the relative position For attitude loop, time is the cost functional to be optimized to perform fast reorientations with wheels to capture the imaging spots in specified time and also to acquire the desired thruster orientation at the time specified for Orbit correction.

15 LQR Based Controller The cost functional for the LQR controller is given by 1 J = x T Q(t)x u T R(t)u dt x T (tf 2 )Qtx(tf ) Where, Q(t) is a positive semidefinite state weighting matrix, R(t) is a positive definite control weighting matrix and Q t is positive definite terminal weighting matrix. The objective is to find the feedback gain K which minimizes the cost functional J. The optimum feedback gain is given by K = R 1 B T P(t) The matrix P is computed by solving the continuous time Riccati equation P &(t) + A T P(t) + P(t)A P(t)BR 1 B T P(t) + Q= 0

16 Block Diagram of Satellite Formation Control Loop (Coupled Position and Attitude) Desired Trajectory in ECI Frame Controller (LQR based) Control Actuator (Thruster & Wheels) Spacecraft Translational and Rotational Dynamics Combined Attitude and Position Estimator IRAP GPS Star Tracker

17 Effect of Initial Orbit Element Separation on S/C Relative Position and Velocity (Open Loop)

18 Effect of Initial Separation in a and e on Open Loop Position and Velocity

19 Effect of Initial Separation in ω and Ω on Open Loop Position and Velocity

20 Effect of Initial Separation in i and M (or θ) on Open Loop Position and Velocity

21 Closed Loop Control Performance

22 Dynamics Parameters Parameter S/C Mass Maximum Thrust Orbit Altitude Inclination Right Ascension Maximum Wheel Momentum Maximum Wheel Torque S/C Inertia about Yaw, Roll and Pitch Leader Follower Star Tracker Update time for Gyro GPS update period for IRAP based position Value 80.0 Kg 0.2 N Km ⁰ ⁰ RPM Nm 10, 9, 9.5 Kg-m 2 11, 8, 9 Kg-m ms 10.0 s

23 Simulation Scenario Leader s Orbital Parameters a, e, i,ω,ω, and M at Epoch (start) are (7198 Km, , , , and ) Initially Leader is Nadir Pointing and Follower is away from Leader in position by [-68 m -214 m -1500m ] in radial, along track and cross track direction. This corresponds to relative error of 100 m less in a, 3 % less in e and 0.01 less in i From this the follower is reconfigured to relative position of [0.0, 100 m, -200 m] position and by control and maintained (i.e., zero relative velocity).

24 Initial Conditions & Mission Requirements (Typical simulation case) Around 7000s, the follower is reconfigured to [0.0, -200m, 200m] relative position and zero relative velocity. From to 13000s, Leader is viewing off-track spot by rotating -26 about Roll (side view) and -45 about Pitch (Back View). By control action, the follower also points the same spot as that of Leader Thus we will get stereo view of the spot by Leader and Follower at different look angles

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28 Conclusions Effects of Initial Orbit Element Differences on Position and Velocity is studied A closed loop control scheme for relative position and attitude control has been demonstrated using LQR based controller It is observed that Stereo view of Spots (SPOT IMAGES) is possible with Leader-Follower relative Control Modeling of Navigation System Errors and Tuning of Kalman Filters form the future part of the work

29 References 1. DL He, XB Cao, J Ma and,px Liu, Impulsive Formation Control Approach Based on Relative Eccentricity/Inclination Vector, Journal of Systems Engineering and Electronics, China National Knowledge Infrastructure, Issue 04, Q He and C Han, Dynamics and Control of Satellite Formation Flying Based on Relative Orbit Elements, AIAA Guidance, Navigation and Control Conference and Exhibit, AIAA , Honolulu, Hawaii, August JF Vasconcelos, C Silvestre and P. Oliveira, A Nonlinear GPS/IMU Based Observer for Rigid Body Attitude and Position Estimation, Proceedings of the 47 th IEEE Conference on Decision and Control, Cancun, Mexico, Dec 9 11,2008, pp S D Amico, O Montenbruck, C Arbinger and H Fiedler, Formation Flying Concept for Close Remote Sensing Satellites, AAS , 15 th AAS/AIAA Space Flight Mechanics Conference, Colorado, January, YT Gao, B Xu and YP Lu, The Kinematics and Control of Relative Attitude Pointing for Spacecraft Formation Flying, IEEE Control and Decision Conference, June 2009., pp

30 Thank You

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